Phyllotaxis and related topics

Phyllotaxis, a keystone problem in structural plant biology, has attracted the interdisciplinary interest of biologists, mathematicians, physicists, crystallographers and, more recently, computer scientists for almost two centuries.The continuing interest in phyllotaxis has led to this special issue, bringing together nine papers from authors who responded to our invitation or call for papers from early 2016.We thank the authors for their contributions, which we have the pleasure of presenting here.The papers focus on two aspects of phyllotactic patterning: its universality across a very broad spectrum of plants, and the interplay between the regularity and irregularity of the patterns.Addressing the first aspect, Gola and Banasiak [1] present a systematic survey of phyllotactic patterns in land plants and show that the same pattern types can be found in analogous (not homologous) axial organs present in different plant groups, from bryophytes to angiosperms.The question of what mechanisms, undoubtedly different in their physiological nature, guarantee this convergence, is still open.Gola and Banasiak also discuss individual features of specific plant groups: for example, the presence of true main Fibonacci pattern in gametophytic shoots of Bryophytes is questionable, and in sporophytic shoots of Lycophytes it is rare.The universality of phyllotaxis is further illustrated by the observation of Fibonacci phyllotaxis in brown algae, a lineage of phototrophic eukaryotes phylogenetically very distant from land plants.In the study of the architecture of the highly organized thallus of the invasive brown alga Sargassum muticum, Peaucelle and Couder [2] emphasize the generic character of self-organizing processes leading to Fibonacci-related spirals, but also point out that, although auxin presence in lower phototrophic eukaryots has been confirmed, the physiological mechanisms of phyllotaxis in these phototrophs are still unknown.For a long time, the regularity of (real or idealized) phyllotactic patterns, and their relation to the golden section and Fibonacci numbers, have been at the center of studies of phyllotaxis.In this context, Okabe [3] gives a historical account of ideas emphasizing the importance of a constant divergence angle -especially, the golden angle -to the geometry of phyllotactic arrangements.In particular, he brings to light the largely forgotten German-language contributions from the first decades of the twentieth century by Hirmer.More recently, however, it is the irregularity observed in real phyllotactic patterns that has gained increasing attention.Combining the interest in both regular and irregular patterns, Golé, Dumais, and Douady [4] raise two fundamental questions: (i) what is the developmental origin of Fibonacci patterns, and (ii) what are the origins and manifestations of departures from this idealized pattern.Extending a line of geometric models going back to van Iterson [5] and Mitchison [6], and building upon the analogy between the dislocations in crystals and the "imperfections" of phyllotactic patterns introduced by Zagórska-Marek [7], Golé et al. present a model focused on the front of an emerging pattern, where new primordia are being added.This focus emphasizes local configurations and interactions between primordia, while deemphasizing other notions traditionally used in the description of phyllotactic patterns, in particular that of the divergence angle (as Okabe [3] points out, the progress in phyllotactic research does not follow a straight line).Using a series of

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